\(\int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx\) [1309]

Optimal result

Integrand size = 25, antiderivative size = 223 \[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a+b) (c-i d) f (1+m)}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) (i c-d) f (1+m)}+\frac {d^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (1+m)} \]

[Out]

1/2*hypergeom([1, 1+m],[2+m],(a+b*tan(f*x+e))/(a-I*b))*(a+b*tan(f*x+e))^(1+m)/(I*a+b)/(c-I*d)/f/(1+m)-1/2*hype
rgeom([1, 1+m],[2+m],(a+b*tan(f*x+e))/(a+I*b))*(a+b*tan(f*x+e))^(1+m)/(I*a-b)/(c+I*d)/f/(1+m)+d^2*hypergeom([1
, 1+m],[2+m],-d*(a+b*tan(f*x+e))/(-a*d+b*c))*(a+b*tan(f*x+e))^(1+m)/(-a*d+b*c)/(c^2+d^2)/f/(1+m)

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3655, 3620, 3618, 70, 3715} \[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\frac {d^2 (a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right )}{f (m+1) \left (c^2+d^2\right ) (b c-a d)}+\frac {(a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (b+i a) (c-i d)}-\frac {(a+b \tan (e+f x))^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b) (-d+i c)} \]

[In]

Int[(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x]),x]

[Out]

(Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m))/(2*(I*a + b)
*(c - I*d)*f*(1 + m)) - (Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x
])^(1 + m))/(2*(a + I*b)*(I*c - d)*f*(1 + m)) + (d^2*Hypergeometric2F1[1, 1 + m, 2 + m, -((d*(a + b*Tan[e + f*
x]))/(b*c - a*d))]*(a + b*Tan[e + f*x])^(1 + m))/((b*c - a*d)*(c^2 + d^2)*f*(1 + m))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3620

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3655

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/
(c^2 + d^2), Int[(a + b*Tan[e + f*x])^m*(c - d*Tan[e + f*x]), x], x] + Dist[d^2/(c^2 + d^2), Int[(a + b*Tan[e
+ f*x])^m*((1 + Tan[e + f*x]^2)/(c + d*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c -
a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3715

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+b \tan (e+f x))^m (c-d \tan (e+f x)) \, dx}{c^2+d^2}+\frac {d^2 \int \frac {(a+b \tan (e+f x))^m \left (1+\tan ^2(e+f x)\right )}{c+d \tan (e+f x)} \, dx}{c^2+d^2} \\ & = \frac {\int (1+i \tan (e+f x)) (a+b \tan (e+f x))^m \, dx}{2 (c-i d)}+\frac {\int (1-i \tan (e+f x)) (a+b \tan (e+f x))^m \, dx}{2 (c+i d)}+\frac {d^2 \text {Subst}\left (\int \frac {(a+b x)^m}{c+d x} \, dx,x,\tan (e+f x)\right )}{\left (c^2+d^2\right ) f} \\ & = \frac {d^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (1+m)}+\frac {\text {Subst}\left (\int \frac {(a+i b x)^m}{-1+x} \, dx,x,-i \tan (e+f x)\right )}{2 (i c-d) f}-\frac {\text {Subst}\left (\int \frac {(a-i b x)^m}{-1+x} \, dx,x,i \tan (e+f x)\right )}{2 (i c+d) f} \\ & = \frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a+b) (c-i d) f (1+m)}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a+i b) (i c-d) f (1+m)}+\frac {d^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\frac {\left (\frac {\operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a-i b}\right )}{(i a+b) (c-i d)}+\frac {i \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {a+b \tan (e+f x)}{a+i b}\right )}{(a+i b) (c+i d)}-\frac {2 d^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {d (a+b \tan (e+f x))}{-b c+a d}\right )}{(-b c+a d) \left (c^2+d^2\right )}\right ) (a+b \tan (e+f x))^{1+m}}{2 f (1+m)} \]

[In]

Integrate[(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x]),x]

[Out]

((Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a - I*b)]/((I*a + b)*(c - I*d)) + (I*Hypergeometric
2F1[1, 1 + m, 2 + m, (a + b*Tan[e + f*x])/(a + I*b)])/((a + I*b)*(c + I*d)) - (2*d^2*Hypergeometric2F1[1, 1 +
m, 2 + m, (d*(a + b*Tan[e + f*x]))/(-(b*c) + a*d)])/((-(b*c) + a*d)*(c^2 + d^2)))*(a + b*Tan[e + f*x])^(1 + m)
)/(2*f*(1 + m))

Maple [F]

\[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{m}}{c +d \tan \left (f x +e \right )}d x\]

[In]

int((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e)),x)

[Out]

int((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e)),x)

Fricas [F]

\[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{d \tan \left (f x + e\right ) + c} \,d x } \]

[In]

integrate((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e)),x, algorithm="fricas")

[Out]

integral((b*tan(f*x + e) + a)^m/(d*tan(f*x + e) + c), x)

Sympy [F]

\[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{m}}{c + d \tan {\left (e + f x \right )}}\, dx \]

[In]

integrate((a+b*tan(f*x+e))**m/(c+d*tan(f*x+e)),x)

[Out]

Integral((a + b*tan(e + f*x))**m/(c + d*tan(e + f*x)), x)

Maxima [F]

\[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{d \tan \left (f x + e\right ) + c} \,d x } \]

[In]

integrate((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e)),x, algorithm="maxima")

[Out]

integrate((b*tan(f*x + e) + a)^m/(d*tan(f*x + e) + c), x)

Giac [F]

\[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{d \tan \left (f x + e\right ) + c} \,d x } \]

[In]

integrate((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e)),x, algorithm="giac")

[Out]

integrate((b*tan(f*x + e) + a)^m/(d*tan(f*x + e) + c), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m}{c+d\,\mathrm {tan}\left (e+f\,x\right )} \,d x \]

[In]

int((a + b*tan(e + f*x))^m/(c + d*tan(e + f*x)),x)

[Out]

int((a + b*tan(e + f*x))^m/(c + d*tan(e + f*x)), x)